Fiche de révision : Introduction aux statistiques et mesures essentielles

📋 Plan du Cours

1. Utilité et applications fondamentales des statistiques
2. Niveaux de mesure des variables : nominal, ordinal, discret et continu
3. Organisation et structuration des données dans les tableaux
4. Mesures de tendance centrale : mode, médiane et moyenne
5. Effet de la distribution asymétrique (skewness) sur les mesures centrales
6. Mesures de dispersion : étendue, intervalle interquartile, variance et écart-type
7. Calcul détaillé de la variance et correction de Bessel (n-1)
8. Interprétation et propriétés de l’écart-type comme mesure de dispersion
9. Introduction au logiciel statistique R : installation, organisation des fichiers et projets
10. Gestion des erreurs et bonnes pratiques pour l’utilisation de R en analyse statistique
11. Compétences visées : calcul, interprétation des statistiques univariées et bivariées, et usage de logiciels
12. Résumé des notions clés : niveaux de mesure, mesures centrales, dispersion et introduction à R

📖 1. Utilité et applications fondamentales des statistiques

🔑 Notions clés & Définitions

• Central Tendency : A statistical concept that identifies the typical or central value of a variable, summarizing important information about its distribution.
• Uncertainty Across Disciplines Statistics gives : The role of statistics as a universal language that provides a vocabulary to communicate insights and uncertainty from data across various fields.
• Vocabulary to communicate insights from : A set of statistical terms and concepts that enable clear communication of findings and uncertainty derived from data.

📝 Points essentiels

• Measures of central tendency summarize important information about a variable by identifying its typical or central value.
• 2 Course Setup 3 Levels of Measurement 4 Measures of Central Tendency 5 Measures of Dispersion 6 Working with R 6 / 73 Course Setup 7 / 73 Workgroup Instructors Pouria Mirelmi, MSc Christofer Talvitie, MSc Duygu Uysal, MSc Raafat Shamieh, MSc Benjamin Kester, MSc Amber Lauwers, MSc 8 / 73 Learning Outcomes In short: Statistical Literacy!

💡 À retenir

Measures of central tendency summarize important information about a variable by identifying its typical or central value.

📖 2. Niveaux de mesure des variables : nominal, ordinal, discret et continu

🔑 Notions clés & Définitions

• Nominal : A measurement level characterized by two or more mutually exclusive categories without any natural ordering, where no arithmetic operations such as addition or subtraction are possible.
• Ordinal : A measurement level with a clear rank ordering of values, where the distances between values are not necessarily equal.

📝 Points essentiels

• Nominal variables include examples such as eye color, marital status, political preference, and favorite football club.
• Ordinal variables include examples such as education level and degree of agreement on a Likert scale.
• Nominal variables do not allow arithmetic operations, while ordinal variables have a clear ordering but unequal distances between values.
• 37 / 73 Measures of Central Tendency 38 / 73 Measures of Central Tendency: Overview Measurement Level Measure of Central Tendency Nominal Mode Ordinal Mode + Median Interval-ratio Mode + Median + Mean 39 / 73 Mode The mode is the most frequent value.

💡 À retenir

Nominal variables include examples such as eye color, marital status, political preference, and favorite football club.

📖 3. Organisation et structuration des données dans les tableaux

🔑 Notions clés & Définitions

• Extended with many : Capacité d’un logiciel statistique à être complété par de nombreux packages additionnels pour étendre ses fonctionnalités.
• Inference for Numerical Data : Domaine d’étude consacré à l’inférence statistique sur des données numériques, incluant la comparaison de moyennes, le test t, la taille d’effet de Cohen et l’ANOVA.
• Offered on a Pay-What- You-Want : Forme de diffusion d’un livre permettant au lecteur de choisir le montant à payer, avec une version gratuite disponible.

📝 Points essentiels

• Le livre OpenIntro statistics est proposé selon un modèle Pay-What-You-Want, incluant une version gratuite.
• 4 / 73 What can you do with it?

💡 À retenir

Le livre OpenIntro statistics est proposé selon un modèle Pay-What-You-Want, incluant une version gratuite.

📖 4. Mesures de tendance centrale : mode, médiane et moyenne

🔑 Notions clés & Définitions

• Mode : Mesure de tendance centrale correspondant à la valeur la plus fréquente dans une série de données.
• Should I learn statistics : Question traitant de l'importance d'apprendre les statistiques à l'ère de l'intelligence artificielle, soulignant que même si l'IA peut réaliser l'analyse des données, l'interprétation, la vérification des hypothèses et la prévention des usages trompeurs nécessitent une compréhension humaine.
• Refer adequately to any literature : Consigne imposant de citer correctement toute littérature utilisée lors de la réalisation des travaux et exercices.

📝 Points essentiels

• Even if AI fully does the data analysis for you… Economics: If data analysis becomes cheaper, it will be more common Interpretation requires understanding Assumptions are not automatically verified Data can be used to mislead Productivity gains are the highest for experts Living in the age of AI requires data literacy - more than ever!
• Refer adequately to any literature you use.

💡 À retenir

Even if AI fully does the data analysis for you… Economics: If data analysis becomes cheaper, it will be more common Interpretation requires understanding Assumptions are not automatically verified Data can be used to mislead Productivity gains are the highest for experts Living in the age of AI requires data literacy - more than ever!

📖 5. Effet de la distribution asymétrique (skewness) sur les mesures centrales

🔑 Notions clés & Définitions

• Exam Grades : A variable representing the score obtained on the Statistics I exam, which can vary between students and over time.
• Projects in RStudio : A structured folder in RStudio where scripts, RMarkdown files, and data files are stored together to organize quantitative analysis work.

📝 Points essentiels

• The exam grade is considered a variable that can be analyzed using data from previous years to understand likely outcomes and their uncertainty.
• In RStudio projects, scripts, RMarkdown files, and data files should be placed together within the project folder.

💡 À retenir

The exam grade is considered a variable that can be analyzed using data from previous years to understand likely outcomes and their uncertainty.

📖 6. Mesures de dispersion : étendue, intervalle interquartile, variance et écart-type

🔑 Notions clés & Définitions

• Interquartile Range : A measure of dispersion calculated as the difference between the third quartile (Q3) and the first quartile (Q1), representing the spread of the middle 50% of the data.
• Variance and Standard Deviation : Measures of dispersion that quantify how values are spread around the mean; variance is the average of squared deviations from the mean, and standard deviation is the square root of the variance, restoring the measure to the original data scale.

📝 Points essentiels

• Range is calculated as the difference between the maximum and minimum values.
• Interquartile range is calculated as Q3 minus Q1.
• 5 60 / 73 Standard Deviation By taking the square root of the variance, we bring the measure of dispersion back to the same scale as the original data.

💡 À retenir

Interquartile range is calculated as Q3 minus Q1.

📖 7. Calcul détaillé de la variance et correction de Bessel (n-1)

🔑 Notions clés & Définitions

• Squared Differences : The squared deviations obtained by taking each value’s difference from the mean and squaring it; these are summed to form the sum of squares.
• Total Deviation : The sum of deviations from the mean, which is always equal to 0.

📝 Points essentiels

• 49 / 73 Calculating Variance (1) 50 / 73 Calculating Variance (2) 51 / 73 Calculating Variance (3) 52 / 73 Calculating Variance (4) 53 / 73 Calculating Variance (5) 54 / 73 Calculating Variance (6) 55 / 73 Calculating Variance (7) 56 / 73 Total Deviation and Squared Differences The total deviation is always equal to 0.
• The sum of the squared differences (sum of squared errors or simply sum of squares) is calculated as: SS = n ∑ i=1 (xi − ¯x)2 57 / 73 Squared Differences 58 / 73 Variance The larger , the larger the sum of squares.

💡 À retenir

49 / 73 Calculating Variance (1) 50 / 73 Calculating Variance (2) 51 / 73 Calculating Variance (3) 52 / 73 Calculating Variance (4) 53 / 73 Calculating Variance (5) 54 / 73 Calculating Variance (6) 55 / 73 Calculating Variance (7) 56 / 73 Total Deviation and Squared Differences The total deviation is always equal to 0.

📖 8. Interprétation et propriétés de l’écart-type comme mesure de dispersion

🔑 Notions clés & Définitions

• Example : How many cups of coffee or tea do you drink per day?

📝 Points essentiels

• 41 / 73 Median (Even Number of Values) If you have an even number of values, you can (when working with numeric values) calculate the median as the average of the two middle values: Arrange all values in a row: 1 5 7 10 The median is then: 5+7 2 = 6 42 / 73 Mean The mean is the sum of all values divided by the number of values.
• 37 / 73 Measures of Central Tendency 38 / 73 Measures of Central Tendency: Overview Measurement Level Measure of Central Tendency Nominal Mode Ordinal Mode + Median Interval-ratio Mode + Median + Mean 39 / 73 Mode The mode is the most frequent value.

💡 À retenir

The notes link central tendency and dispersion to the variable grades$grade_rounded, where the median is 8 and the mean is 7.1. They also state that a larger standard deviation means greater dispersion around the mean.

📖 9. Introduction au logiciel statistique R : installation, organisation des fichiers et projets

🔑 Notions clés & Définitions

• Calculating Variance : A measure of dispersion for interval-ratio data calculated by summing the squared differences between each value and the mean, then dividing by n-1 when using a sample.

📝 Points essentiels

• Variance and standard deviation are the measures of dispersion used for interval-ratio data.
• The range is calculated as the maximum value minus the minimum value, with an example calculation of 6 - 1 = 5.
• The interquartile range is calculated as Q3 minus Q1, with an example calculation of 5 - 1 = 4.

💡 À retenir

The interquartile range is calculated as Q3 minus Q1, with an example calculation of 5 - 1 = 4.

📖 10. Gestion des erreurs et bonnes pratiques pour l’utilisation de R en analyse statistique

🔑 Notions clés & Définitions

• Standard deviation : N n − 1 s2 = SS n − 1 = ∑n i=1(xi − ¯x)2 n − 1 59 / 73 Variance s2 = SS n − 1 = 1166 5 − 1 = 1166 4

📝 Points essentiels

• The course states that the total deviation is always equal to 0, so it is not useful for measuring dispersion.
• The sum of squared errors is divided by n-1 because the course often works with samples and this gives more accurate results, described as Bessel’s correction.
• The variance formula shown is s² = SS / (n - 1).
• The standard deviation is the square root of the variance, written as s = √s² = √(SS / (n - 1)).

💡 À retenir

The standard deviation is the square root of the variance, written as s = √s² = √(SS / (n - 1)).

📖 11. Compétences visées : calcul, interprétation des statistiques univariées et bivariées, et usage de logiciels

🔑 Notions clés & Définitions

• Calcul : Processus d'exécution d'opérations mathématiques sur des données pour obtenir des résultats numériques, comme le calcul de la variance et de l'écart-type à partir de la somme des carrés et de l'effectif.

📝 Points essentiels

• 68 / 73 Projects in RStudio (2) 69 / 73 Projects in RStudio (3) Place scripts, RMarkdown files and data files in the project folder.
• (In Windows Explorer or Finder on Mac) 67 / 73 Projects in RStudio Use a Project in RStudio so that all related files are easy to find.

💡 À retenir

68 / 73 Projects in RStudio (2) 69 / 73 Projects in RStudio (3) Place scripts, RMarkdown files and data files in the project folder.

📖 12. Résumé des notions clés : niveaux de mesure, mesures centrales, dispersion et introduction à R

🔑 Notions clés & Définitions

📝 Points essentiels

• 68 / 73 Projects in RStudio (2) 69 / 73 Projects in RStudio (3) Place scripts, RMarkdown files and data files in the project folder.

💡 À retenir

Ce résumé relie les niveaux de mesure, les mesures de tendance centrale et de dispersion, et introduit l’organisation des fichiers dans R via les projets pour faciliter le travail et la gestion des erreurs, tout en rappelant que les travaux notés doivent être réalisés individuellement.

🧩 Compléments de couverture

1. Today: Summarizing important information about a variable level of measurement central tendency dispersion Statistics is universal
2. Describing reality Discovering patterns Investigating (causal) relationships Working with samples 5 / 73 Overview 1 Why Statistics? 2 Course Setup 3 Levels of Measurement 4 Measures of Central Tendency 5 Measures of Dispersion 6 Working wit
3. Objective 2: Students can calculate and report, as well as critically evaluate others’ use and interpretation of, a number of bivariate statistical analyses
4. Data: Comparing Means T-test, Cohen’s D 7 Inference for Numerical Data: Statistical Power and ANOVA ANOVA χ2 10 / 73 Textbook Diez, D
5. Pay-What- You-Want basis, including a free version
6. 14 / 73 Brightspace Announcements Slides Instructional videos and software guides Important documents (reporting guidelines, formula sheets, FAQ) Quizzes 15 / 73 What do you need to do each week? Read textbook chapters (3-5 hours) Attend le
7. Total: 10-18 hours per week
8. 18 / 73 Why should I learn statistics in the age of AI? Even if AI fully does the data analysis for you… Economics: If data analysis becomes cheaper, it will be more common Interpretation requires understanding Assumptions are not automatic
9. Study: Read the book, watch the videos, attend the lecture and workgroup every week 2
10. Exam Grades: Expectations and the Past 24 / 73 Exam Grades: Expectations and the Past The exam grade is a variable
11. Examples: Eye color (brown, blue, green, …) Marital status (married, unmarried, divorced, widowed, …) Political preference (Labour, CDA, VVD, …) Favorite football club No arithmetic operations possible (subtract, add, etc
12. Examples: Weight (in kilograms or pounds) Time (measured in seconds, minutes, or hours) Blood pressure Body temperature 30 / 73 Discrete Only certain, countable values are possible (usually whole numbers) Examples: Number of pets Exam point
13. Discrete Interval-ratio: only countable (whole) values Number of pets Exam points 32 / 73 Organizing Data Variables in the columns
14. Non-binary <NA> A few times 6 6 0 Male <NA> A few times 7 7 0 Female <NA> A few times 8 8 0 Female <NA> A few times 9 9 1 Male High Regularly 10 10 0 Female High A few times 33 / 73 Nominal Variable: Gender 34 / 73 Ordinal Variable: Compute
15. er Skills 35 / 73 Interval-ratio Variable: Exam Grades 36 / 73 Skewness Variable Grade is not symmetrically distributed, but skewed: a long tail on the left side.
16. 1. 41 / 73 Median (Even Number of Values) If you have an even number of values, you can (when working with numeric values) calculate the median as the average of the two middle values: Arrange all values in a row: 1 5 7 10 The median is the
17. Exam Grades Mode DescTools::Mode(grades$grade_rounded)1 [1] 8
18. Dispersion - Overview Measurement Level Measure of Dispersion Nominal — Ordinal Range, Interquartile Range Interval-ratio Variance or Standard Deviation 47 / 73 Range and Interquartile Range Range 1 1 1 3 4 4 5 6 6 Range = maximum - minimum
19. 1. Not so useful a measure for dispersion
20. That’s why we divide the sum of squared errors by : Why n-1? We often work with samples and by using n-1 we get more accurate results (Bessel’s correction). Watch the video on Brightspace for more explanation. n n − 1 s2 = SS n − 1 = ∑n i=1
21. But: getting started is often a bit more difficult than ‘point and click’ 66 / 73 Organize Your Files Use a folder for all files in this course with subfolders within it
22. Example: a project for all the exercises for this course
23. Note: in R you can often do things in multiple ways
24. n) Skewness: with skewed distribution the mean differs from the median; outliers pull the mean along Dispersion: range, interquartile range (IQR), variance ( ) and standard deviation ( ) R: create projects, load data, fix error messages s2

📊 Tableaux de Synthèse

Niveaux de mesure et mesures associées

Dispersion et calculs

⚠️ Pièges & Confusions Fréquentes

1. Confondre nominal et ordinal : le nominal n’a pas d’ordre naturel, l’ordinal a un rang clair.
2. Utiliser la moyenne pour une variable nominale : le cours associe le mode au niveau nominal.
3. Oublier que la médiane pour un nombre pair de valeurs est la moyenne des deux valeurs centrales.
4. Interpréter l’étendue comme une mesure des valeurs centrales alors qu’elle mesure la dispersion globale.
5. Confondre variance et écart-type : l’écart-type est la racine carrée de la variance.
6. Oublier la correction de Bessel : la variance d’échantillon est divisée par n - 1.
7. Croire que la déviation totale mesure la dispersion : le total des écarts à la moyenne est toujours 0.

✅ Checklist Examen

1. Définir l’utilité des statistiques comme langage pour communiquer des informations et de l’incertitude.
2. Distinguer nominal et ordinal par la présence ou non d’un ordre naturel.
3. Associer le mode au niveau nominal.
4. Associer mode et médiane au niveau ordinal.
5. Associer mode, médiane et moyenne aux données interval-ratio.
6. Calculer l’étendue comme maximum moins minimum.
7. Calculer l’intervalle interquartile comme Q3 moins Q1.
8. Expliquer la variance comme somme des écarts au carré divisée par n - 1 pour un échantillon.
9. Expliquer que l’écart-type est la racine carrée de la variance.
10. Relier l’asymétrie à un écart entre moyenne et médiane.
11. Retenir que la somme des écarts à la moyenne est toujours égale à 0.
12. Savoir qu’en R il faut organiser ses fichiers en dossiers et projets et gérer les messages d’erreur.